What is the formula for the distance between two points?

The formula for the distance between two points in a two-dimensional Cartesian coordinate system is determined by the Pythagorean theorem. When calculating the distance between the points ((x_1, y_1)) and ((x_2, y_2)), you first find the differences in their respective coordinates, which are ((x_2 - x_1)) and ((y_2 - y_1)).

By squaring these differences, the expression ((x_2 - x_1)^2 + (y_2 - y_1)^2) gives you the sum of the squares of the horizontal and vertical distances between the points. To find the straight-line distance, you then take the square root of that sum, resulting in the formula (\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}).

Thus, the formula represented in the correct choice captures the essence of finding the direct distance between two points in the plane by integrating both coordinates' differences while adhering to the Pythagorean principle.